# Are these two definitions of limit actually equivalent?

I've seen the following definition of limits in the given books:

Laczkovich/Sós: Real Analysis

Let $$f$$ be defined on an open interval containing $$a$$, excluding perhaps $$a$$ itself. The limit of $$f$$ at $$a$$ exists and has value $$b$$ if for all $$\epsilon>0$$, there exists a $$\delta >0$$ such that

$$|f(x)-b|<\epsilon, \text{ if } 0<|x-a|<\delta$$

On the other hand:

Zorich: Mathematical Analysis 1

On a function $$f:E\mapsto \Bbb{R}$$.

$$\forall \epsilon>0\;\; \exists \delta >0 \;\; \forall x\in E \;\; (0<|x-a|<\delta \implies |f(x)-A|<\epsilon)$$

And this sides up with Spivak's definition on his Calculus and many other books I've seen. In the second definition, the inclusion of $$\forall x\in E$$ seems to tell me something very different. I tried to prove the inexistence of a limit in the Dirichlet function yersterday with the first definition and couldn't but when I read Zorich's definition, it was pretty straightforward so, are both definitions actually equivalent? I guess in the first definition, the authors expect us to understand $$\forall x \in E$$ by pointing to a figure in the page perhaps.

• The set $E$ in the second definition is the open interval in the first definition. The point is that $f(x)$ only makes sense if $x$ is in the domain of $f$. In your first definition, this is implicit; in the second definition, it is made explicit. – Xander Henderson Mar 4 '19 at 0:28
• ^^ This is better than either current answer, IMHO. – The Count Mar 4 '19 at 0:53

They are exactly the same definition. In the second the domain $$E$$ is made explicit, while in the first it's not. Think about where are the $$x$$ coming from in the first definition.
They are the same definition, with $$b=A$$, and $$E$$ being the open interval as defined above.